Details
Unified Theory for Fractional and Entire Differential Operators
An Approach via Differential Quadruplets and Boundary Restriction OperatorsFrontiers in Mathematics
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58,84 € |
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| Verlag: | Birkhäuser |
| Format: | |
| Veröffentl.: | 27.06.2024 |
| ISBN/EAN: | 9783031583568 |
| Sprache: | englisch |
Dieses eBook enthält ein Wasserzeichen.
Beschreibungen
<p>This monograph proposes a unified theory of the calculus of fractional and standard derivatives by means of an abstract operator-theoretic approach. By highlighting the axiomatic properties shared by standard derivatives, Riemann-Liouville and Caputo derivatives, the author introduces two new classes of objects. The first class concerns differential triplets and differential quadruplets; the second concerns boundary restriction operators. Instances of boundary restriction operators can be generalized fractional differential operators supplemented with homogeneous boundary conditions. The analysis of these operators comprises:</p>
<ul>
<li>The computation of adjoint operators;</li>
<li>The definition of abstract boundary values;</li>
<li>The solvability of equations supplemented with inhomogeneous abstract linear boundary conditions;</li>
<li>The analysis of fractional inhomogeneous Dirichlet Problems.</li>
</ul>
<p>As a result of this approach, two striking consequences are highlighted: Riemann-Liouville and Caputo operators appear to differ only by their boundary conditions; and the boundary values of functions in the domain of fractional operators are closely related to their kernel.</p>
<p><em>Unified Theory for Fractional and Entire Differential Operators</em> will appeal to researchers in analysis and those who work with fractional derivatives. It is mostly self-contained, covering the necessary background in functional analysis and fractional calculus.</p>
<ul>
<li>The computation of adjoint operators;</li>
<li>The definition of abstract boundary values;</li>
<li>The solvability of equations supplemented with inhomogeneous abstract linear boundary conditions;</li>
<li>The analysis of fractional inhomogeneous Dirichlet Problems.</li>
</ul>
<p>As a result of this approach, two striking consequences are highlighted: Riemann-Liouville and Caputo operators appear to differ only by their boundary conditions; and the boundary values of functions in the domain of fractional operators are closely related to their kernel.</p>
<p><em>Unified Theory for Fractional and Entire Differential Operators</em> will appeal to researchers in analysis and those who work with fractional derivatives. It is mostly self-contained, covering the necessary background in functional analysis and fractional calculus.</p>
<p>Introduction.- Background on Functional Analysis.- Background on Fractional Calculus.- Differential Triplets on Hilbert Spaces.- Differential Quadruplets on Banach Spaces.- Fractional Differential Triplets and Quadruplets on Lebesgue Spaces.- Endogenous Boundary Value Problems.- Abstract and Fractional Laplace Operators.</p>
<p>This monograph proposes a unified theory of the calculus of fractional and standard derivatives by means of an abstract operator-theoretic approach. By highlighting the axiomatic properties shared by standard derivatives, Riemann-Liouville and Caputo derivatives, the author introduces two new classes of objects. The first class concerns differential triplets and differential quadruplets; the second concerns boundary restriction operators. Instances of boundary restriction operators can be generalized fractional differential operators supplemented with homogeneous boundary conditions. The analysis of these operators comprises:</p>
<ul>
<li>The computation of adjoint operators;</li>
<li>The definition of abstract boundary values;</li>
<li>The solvability of equations supplemented with inhomogeneous abstract linear boundary conditions;</li>
<li>The analysis of fractional inhomogeneous Dirichlet Problems.</li>
</ul>
<p>As a result of this approach, two striking consequences are highlighted: Riemann-Liouville and Caputo operators appear to differ only by their boundary conditions; and the boundary values of functions in the domain of fractional operators are closely related to their kernel.</p>
<p><em>Unified Theory for Fractional and Entire Differential Operators</em> will appeal to researchers in analysis and those who work with fractional derivatives. It is mostly self-contained, covering the necessary background in functional analysis and fractional calculus.</p>
<ul>
<li>The computation of adjoint operators;</li>
<li>The definition of abstract boundary values;</li>
<li>The solvability of equations supplemented with inhomogeneous abstract linear boundary conditions;</li>
<li>The analysis of fractional inhomogeneous Dirichlet Problems.</li>
</ul>
<p>As a result of this approach, two striking consequences are highlighted: Riemann-Liouville and Caputo operators appear to differ only by their boundary conditions; and the boundary values of functions in the domain of fractional operators are closely related to their kernel.</p>
<p><em>Unified Theory for Fractional and Entire Differential Operators</em> will appeal to researchers in analysis and those who work with fractional derivatives. It is mostly self-contained, covering the necessary background in functional analysis and fractional calculus.</p>
Proposes a unified theory of fractional and entire derivatives Focuses on the solvability of linear boundary value problems based on usual and fractional differential operators Establishes the theory utilizing a simple framework that makes it more accessible to researchers
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